Codes over Trees Lev Yohananov and Eitan Yaakobi Technion – Israel Institute of Technology 2020 IEEE International Symposium on Information Theory
Motivation • Trees and their properties are very beneficial in numerous applications. • In biology. • In chemistry. • In programming languages. • In cyber applications. • etc. • In coding theory , a novel family of codes is presented. 1
Trees • A finite undirected tree over 𝑜 nodes is a connected undirected graph with 𝑜 − 1 edges. • 𝑈 𝑜 : the set of all trees over 𝑜 nodes. 9 1 2 3 4 5 6 7 8 • By Cayley ’ s formula 1 it holds that |𝑈 𝑜 | = 𝑜 𝑜−2 . 1 M. Aigner and G. M. Ziegler, Proofs from THE BOOK, pp. 141 – 146,Springer-Verlag, New York, 1998. 2
Codes (over Trees) • 𝐷 𝑈 : a code (over trees) denoted by 𝑜, 𝑁 𝑈 , such that • 𝑜: the number of nodes in a tree. • 𝑁: the size of 𝐷 𝑈 . • Example 𝑜 = 5, 𝑁 = 8 4 4 1 4 3 0 1 3 1 0 3 2 0 2 3 0 2 2 4 1 4 4 4 4 3 1 0 3 1 0 3 2 1 2 3 1 2 2 0 0 3
Edge Erasure • An edge erasure is a removal of an edge from a tree. 4
Tree Distance • Given two trees 𝑈 1 = (𝑊 𝑜 , 𝐹 1 ) and 𝑈 2 = 𝑊 𝑜 , 𝐹 2 . • 𝑒 𝑈 𝑈 1 , 𝑈 2 : the tree distance (or distance ) between 𝑈 1 and 𝑈 2 is 𝑒 𝑈 𝑈 1 , 𝑈 2 = 𝑜 − 1 − |𝐹 1 ∩ 𝐹 2 |. 9 9 1 2 3 4 5 1 2 3 4 5 6 7 6 7 8 8 𝑒 𝑈 𝑈 1 , 𝑈 2 = 8 − 7 = 1. • This distance is a metric. 5
Codes with Minimal Distance 𝑒 • 𝑜, 𝑁, 𝑒 𝑈 : a code over trees of minimal distance 𝒆. • 𝑠: the redundancy of the code 𝐷 𝑈 . • 𝑠 = 𝑜 − 2 log 𝑜 − log(𝑁) . • 𝐵 𝑜, 𝑒 : the largest size of a code of distance 𝑒 . • 𝑠 𝑜, 𝑒 : the minimal redundancy of a code of distance 𝑒 . • Theorem: A 𝑜, 𝑁 𝑈 code over trees 𝐷 𝑈 is of tree distance 𝒆 if and only if it can correct any 𝒆 − 𝟐 edges . 6
Forests • An undirected graph that consists of only disjoint union of trees is called a forest . 9 1 2 3 4 5 6 7 8 • ℱ 𝑜, 𝑢 : the set of all forests over 𝑜 nodes with exactly 𝒖 trees . • Note that ℱ(𝑜, 1) = 𝑈(𝑜) . 7
Number of Forest with Exactly 𝑢 Trees • The value of |ℱ 𝑜, 𝑢 | was shown to be 𝑗 𝑢 𝑢 − 1 (𝑢 + 𝑗) 𝑜 − 𝑢 ! 𝐺 𝑜, 𝑢 = 𝑜 𝑢 𝑜 𝑜−𝑢−1 𝑜 𝑗 𝑜 − 𝑢 − 𝑗 ! . 𝑗 2 𝑗=0 J. Moon 2 1970. • Another representation of it 𝑗 𝑢 𝑢 − 1 𝑢 + 𝑗 ! 𝑜 − 1 𝐺 𝑜, 𝑢 = 𝑜 𝑜−𝑢 . 𝑗 𝑜 𝑗 𝑢! 𝑢 − 1 + 𝑗 2 𝑗=0 B. Bollobas 3 1979. 2 J. W. Moon, Counting labeled trees, 1970. 3 B. Bollobas, Graph Theory: An Introductory Course, Springer-Verlag, New York, 1979. 8
Forest Ball of a Tree • 𝒬 𝑈 𝑜, 𝑢 : the forest ball of a tree 𝑈 of radius 𝑢. • Note that 𝒬 𝑈 𝑜, 𝑢 ⊆ ℱ(𝑜, 𝑢 + 1) . 4 4 1 0 2 1 0 2 4 3 3 𝑈 = 𝒬 𝑈 𝑜 = 5, 𝑢 = 1 = Regular 1 0 2 4 4 3 1 0 2 1 0 2 3 3 Note that |𝒬 𝑈 𝑜, 𝑢 | = 𝑜 − 1 𝑢 9
Sphere Packing Bound • Theorem: For all 𝑜 ≥ 1 and 1 ≤ 𝑒 ≤ 𝑜 , it holds that 𝐵 𝑜, 𝑒 ≤ 𝐺(𝑜, 𝑒)/ 𝑜 − 1 𝑒 − 1 . 𝐺 𝑈 2 𝑈 1 𝑒 − 1 10
Corollary • It was also proved 2 that for any fixed 𝑒 , 𝐺 𝑜, 𝑒 1 lim = 2 𝑒−1 𝑒 − 1 ! . 𝑜 𝑜−2 𝑜→∞ • Thus, 𝐵 𝑜, 𝑒 ≤ 𝐺 𝑜, 𝑒 = 𝑃 𝑜 𝑜−1−𝑒 . 𝑜 − 1 𝑒 − 1 2 J. W. Moon, Counting labeled trees, 1970. 11
Results from Sphere Packing Bound 𝑜 • Correcting 𝒐 − 𝟑 erasures : 𝐵 𝑜, 𝑜 − 1 ≤ 2 . • Correcting 𝒐 − 𝟒 erasures : 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜 2 . • Correcting 𝒐 − 𝟓 erasures : 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 3 . 12
The Results of this Work 𝐵 𝑜, 𝑜 − 1 ≤ 𝑜 • Correcting 𝒐 − 𝟑 erasures : 𝐵 𝑜, 𝑜 − 1 = 𝑜/2 . 2 . 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜 2 . • Correcting 𝒐 − 𝟒 erasures : 𝐵 𝑜, 𝑜 − 2 = 𝑜. • Correcting 𝒐 − 𝟓 erasures : 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 2 . 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 3 . • For fixed 𝑒 and 𝑜 ≥ 2𝑒 , Ω 𝑜 𝑜−2𝑒 ≤ 𝐵 𝑜, 𝑒 ≤ 𝑃 𝑜 𝑜−1−𝑒 . 13
Lower Bound on 𝐵(𝑜, 𝑜 − 1) • A line tree 𝑈 : 1 0 2 3 4 • Our code will be constructed from 𝑜/2 line trees as follows: 3 2 2 4 1 3 1 5 0 4 0 6 7 5 7 6 • Thus, this code is a set of 𝑜/2 disjoint Hamiltonian paths, see Lucas 4 . 𝐵 𝑜, 𝑜 − 1 = 𝑜/2 4 E. Lucas, “ Les rondes enfantines, ” Recreations mathematiques, vol. 2, Paris, 1894. 14
Lower Bound on 𝐵(𝑜, 𝑜 − 2) 4 • A star tree 𝑈 : 1 0 2 • Our code will be constructed from 𝑜 star trees as follows: 3 4 4 4 4 0 1 0 2 0 1 2 1 2 0 1 3 2 1 4 2 3 3 3 0 3 • Every two trees have exactly one edge in common. 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜 2 𝐵 𝑜, 𝑜 − 2 ≥ 𝑜 15
Upper Bound on 𝐵(𝑜, 𝑜 − 2) egree = 𝒐 − 𝟐 deg 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜 2 𝑓 1 𝑈 𝐵 𝑜, 𝑜 − 2 ≤ 𝑜 1 𝑓 2 𝑈 2 • Let 𝐻 be a bipartite graph as in example: ⋮ ⋮ 𝑓 𝑜 𝑈 𝑁 2 • Any code of tree distance 𝑜 − 2 yields a bipartite 𝐻 of girth of at least 6 . • Using Reiman’s inequality 5 1958, it is shown that every such bipartite 𝐻 holds 𝑁 ≤ 𝑜. 5 I. Reiman, “Uber ein Problem von K. Zarankiewicz,” Acta mathematica hungarica, vol. 9, issue 3 – 4, pp. 269 – 273, 16 Hungary, Budapest, Sep. 1958.
Upper Bound on 𝐵(𝑜, 𝑜 − 3) 𝐵 𝑜, 𝑜 − 3 = 𝑃(𝑜 2 ) 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 3 • Also done using Reiman’s inequality. (short in time) = Ω(𝑜 2 ). • In the longest version 6 : 𝐵 𝑜, 3𝑜/4 • 𝑜 is a prime number. 6 L. Yohananov and E. Yaakobi, “ Codes over trees, ” arXiv:2001.01791,Jan. 2020. 17
Lower Bound on General 𝐵(𝑜, 𝑒) • Theorem: for fixed 𝑒 and 𝑜 ≥ 2𝑒 it holds 𝐵 𝑜, 𝑒 = 𝛻 𝑜 𝑜−2𝑒 . • Construction: Let (𝑓 1 , 𝑓 2 , … , 𝑓 𝑜 ) be some order of all the edges of 2 the complete graph over 𝒐 nodes . • Each tree 𝑈 will be represented as a characteristic vector of length 𝑜 2 and weight 𝒐 − 𝟐 as in the example: 1 0 2 1 1 1 0 0 0 {0,1} {0,2} {0,3} {1,2} {1,3} {2,3} 3 18
The Proof • A linear binary code of length 𝑂 = 𝑜 2 and Hamming distance 𝐸 = 2𝑒 − 1 can correct at most 𝑒 − 1 substitution. • Corresponding to 𝑒 − 1 edge erasures . • Applying BCH codes , we pay the redundancy of 𝑠 = 𝑒 − 1 log 𝑜 2 + 𝑃 1 = 2 𝑒 − 1 log 𝑜 + 𝑃 1 . • The 2 𝑠 cosets of such code are also ( 𝑜 2 , 𝐿, 2𝑒 − 1) codes. • Thus, by the pigeonhole principle there is a code of cardinality at least 𝑜 𝑜−2 2 2 𝑒−1 log 𝑜 = Ω 𝑜 𝑜−2𝑒 . 19
Tree Balls of Trees 20
Tree Ball of Trees • 𝐶 𝑈 𝑜, 𝑢 : tree ball of trees: 𝒬 𝑈 𝑜, 𝑢 4 4 ⋯ 1 0 2 1 0 2 4 3 3 Not Regular 1 0 2 Regular 4 4 3 ⋯ 1 0 2 1 0 2 |𝐶 𝑈 𝑜, 𝑢 | =? |𝒬 𝑈 𝑜, 𝑢 | = 𝑜 − 1 3 3 𝑢 ⋯ ⋯ 21
Radius One Arbitrary 𝑈 𝑈 𝑈 𝑈 Θ(𝑜 2 ) Average ball size: Θ(𝑜 2.5 ) Θ(𝑜 3 ) Explicit formulas. 22
Radius 𝑢 (fixed) Arbitrary 𝑈 𝑈 𝑈 𝑈 Θ(𝑜 2𝑢 ) Average ball size: Θ(𝑜 2.5𝑢 ) Θ(𝑜 3𝑢 ) Recursive formulas. 23
The Results of this Work • Correcting 𝒐 − 𝟑 erasures : 𝐵 𝑜, 𝑜 − 1 = 𝑜/2 . • Correcting 𝒐 − 𝟒 erasures : 𝐵 𝑜, 𝑜 − 2 = 𝑜. • Correcting 𝒐 − 𝟓 erasures : 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 2 . • For fixed 𝑒 and 𝑜 ≥ 2𝑒 , Ω 𝑜 𝑜−2𝑒 ≤ 𝐵 𝑜, 𝑒 ≤ 𝑃 𝑜 𝑜−1−𝑒 . • Studying tree balls of trees. 24
Conclusions and Future Work • Improve the lower and upper bounds on 𝐵(𝑜, 𝑒) . • Study codes over trees under different metrics such as the tree edit distance. • Study the problem of reconstructing trees based upon several forests in the forest ball of trees. Thank You! 25
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